We define and explore no-upward-crossing (NUC), a condition satisfied by every parameterized family of distributions commonly used in economic applications. Under smoothness assumptions, NUC is equivalent to log-supermodularity of the negative of the derivative of the distribution with respect to the parameter. It is characterized by a natural monotone comparative static and is central in establishing quasi-concavity in a family of decision problems. As an application, we revisit the first-order approach to the moral-hazard problem. NUC simplifies the relevant conditions for the validity of the first-order approach and gives them an economic interpretation. We provide extensive analysis of sufficient conditions for the first-order approach for exponential families.We are grateful to John Conlon, Ian Jewitt, Ohad Kadan, and three anonymous referees for very helpful comments and suggestions that significantly improved the paper.© 2020 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org. https://doi.org/10.3982/TE2937 1 In the moral-hazard context, a weaker version of this condition appears in one result in Jung and Kim (2015a) as sufficient to justify the first-order approach. We discuss this paper further below.2 TP 3 has appeared in the literature on the first-order approach to the moral-hazard problem, most prominently in Jewitt (1988, p. 1182) (where one can find a definition and discussion), and also in Jung and Kim (2015a).